Ballistic ranging methods and systems for inclined shooting

ABSTRACT

A portable system for facilitating inclined shooting of projectile weapons comprises a ranging system, an inclinometer and a processor. The ranging system measures a line-of sight range distance from a vantage point to a target that is elevated or depressed relative to the vantage point, and the inclinometer measures an inclination angle of a line of sight between the vantage point and the target. Based on information from the rangefinder and inclinometer, the processor determines a predicted altitude-compensated inclined shooting (ACIS) trajectory at the line-of sight range distance for a preselected projectile. The ACIS trajectory is based on a bullet path height correction between a bullet path height at a first altitude and a bullet path height at a second altitude, a range distance of the target from the vantage point, and selected meteorological atmospheric information.

CROSS-REFERENCE TO RELATED PATENT APPLICATION

The present patent application is a continuation patent application ofU.S. patent application Ser. No. 12/952,121, by William T. McDonald etal., entitled “Ballistic Ranging Methods and Systems For InclinedShooting,” and filed Nov. 22, 2010, now U.S. Pat. No. 8,172,139, thedisclosure of which is incorporated by reference herein.

BACKGROUND

The subject matter disclosed herein relates to methods and systems forcompensating for ballistic drop and to portable devices (such as variousequipments embodying various target locating and designators)implementing such methods. More particularly, the subject matterdisclosed herein relates to method and system for compensating forballistic drop for inclined shooting and to rangefinders and otherportable devices implementing such methods.

BRIEF DESCRIPTION OF THE DRAWINGS

The subject matter disclosed herein is illustrated by way of example andnot by limitation in the accompanying figures in which like referencenumerals indicate similar elements and in which:

FIG. 1 depicts a schematic diagram of level-fire and inclined-firetrajectories for a projectile;

FIG. 2 depicts a schematic diagram illustrating measurements and factorsin calculating an Equivalent Horizontal Range;

FIG. 3 depicts a flow chart for one exemplary embodiment of a method fordetermining the Equivalent Horizontal Range for accurately aiming aprojectile weapon at an elevated or depressed target located at ainclined line of sight;

FIG. 4 depicts a summary of one exemplary method for calculating atrajectory parameter of bullet path and Equivalent Horizontal Range forbullets;

FIG. 5 depicts a summary of one exemplary method for calculating atrajectory parameter of an arrow path and equivalent horizontal rangefor arrows

FIG. 6 depicts a flow diagram for one exemplary embodiment of operationsand a computational process performed by a master processor forgenerating reference trajectory information for computingAltitude-Compensated Inclined Shooting (ACIS) trajectory information fora selected cartridge according to the subject matter disclosed herein,the ACIS method being an alternative to the equivalent horizontal rangemethod for generating reference trajectory information;

FIG. 7 depicts a flow diagram for one exemplary embodiment of operationsand computational process performed by a device processor for generatingAltitude-Compensated Inclined Shooting (ACIS) trajectory information fora selected cartridge according to the subject matter disclosed herein;

FIG. 8 depicts an exemplary embodiment of a portable handheldrangefinder that generates Attitude-Compensated Inclined Shooting (ACIS)trajectory information for a selected cartridge;

FIG. 9 depicts an enlarged view of an exemplary embodiment of anelectronic display as viewed through an eyepiece of the exemplaryportable handheld rangefinder depicted in FIG. 8;

FIG. 10 depicts an exemplary block diagram for an exemplary embodimentof rangefinder device according to the subject matter disclosed herein;and

FIG. 11 depicts an exemplary embodiment of a telescopic sighting devicefor use with the subject matter disclosed herein.

DETAILED DESCRIPTION

The word “exemplary,” as used herein, means “serving as an example,instance, or illustration.” Any embodiment described herein as“exemplary” is not to be construed as necessarily preferred oradvantageous over other embodiments. Also, six technical terms usedrepeatedly herein require explanation. These terms are “ballistic path,”“bullet path,” “arrow path,” “ballistic path height,” “bullet pathheight,” and “arrow path height.” A projectile flying through the airwithout propulsion follows a ballistic trajectory, which may also becalled a “ballistic path.” Two types of projectiles are addressedherein, bullets from firearms and arrows from bows. Thus, the “ballisticpath” of a bullet is a “bullet path,” and similarly, the “ballisticpath” of an arrow is an “arrow path”. When the word “height” is added toany of these terms (e.g., “bullet path height”), it refers specificallyto the perpendicular distance between the instantaneous position of theprojectile (e.g., the bullet) in flight and the extended line of sightof a shooter through the sighting device on the weapon which launchedthe projectile. The path height is considered positive when theprojectile is above the extended line of sight, and negative when theprojectile is below the extended line of sight.

FIG. 1 depicts a schematic diagram illustrating the effect on thetrajectory of a projectile of the inclination of the line along whichprojectile is fired, cast, or otherwise launched (the “line of initialtrajectory” or, in the case of guns, the “bore line”). For purposes ofillustration, the trajectory curves and angles between various lines inFIG. 1 are greatly exaggerated and are not to scale.

With reference to FIG. 1, a “level fire” trajectory is the path alongwhich a projectile moves when shot at a target T at range R₀ and atsubstantially the same geographic elevation as a vantage point VP of theshooter. The weapon launching the projectile has a line of initialtrajectory (“level-fire bore line”) that is not actually level, butrather is inclined relative to the level-fire line of sight (level-fireLOS) by an elevation angle α. The angle α is quite small, typicallyabout one minute of angle (MoA) for a firearm, and larger (several MoA)for a bow. The level-fire line of sight, which is thereforeapproximately horizontal, begins at a height h above the beginning ofthe bore line. The height h and elevation angle α represent the typicalmounting arrangement of a sighting device (i.e., riflescope, opensights, etc.) on a firearm or an archery sight on a bow. The level-firetrajectory intersects the level-fire line of sight at range R₀ and isknown as the “sighted-in range” or “zero range” or “zeroed-in range”(also referred to herein as zero-range distance R_(Z)) of the weapon andsight combination. The sighted-in range R₀ is typically established byshooting the weapon at a target at a known horizontal reference distanceR₀, such as 100 yards, and adjusting the elevation angle α of theriflescope or other sighting device until projectiles shot by the weaponimpact the target at a point that coincides with the cross hairs orother aiming mark of the riflescope or other sighting device.

An “inclined-fire trajectory” is also depicted in FIG. 1. Theinclined-fire trajectory represents the path along which the sameprojectile travels when aimed at a target that is elevated relative tovantage point VP. The height h and elevation angle α of theinclined-fire line of sight relative to the bore line are the same as inthe level-fire scenario, because there can be no adjustment to thesighting device on the firearm or bow to anticipate the target elevationin the field. The inclined-fire line of sight will be inclined by anangle of inclination θ. As illustrated in FIG. 1, the inclined-firetrajectory crosses the inclined-fire line of sight at a distancesubstantially greater than the sighted-in range R₀. This overshoot isdue to the effect of gravity, which always acts in the verticallydownward direction, regardless of the angle of inclination θ. Theovershoot phenomena and prior methods of correcting for it are discussedin detail by W. T. McDonald in his paper titled “Inclined Fire” (June2003), available from sierrabullets.com. The effects of inclination aretypically even more pronounced in archery than for bullets and arecaused by differences in the initial speed and aerodynamiccharacteristics of the projectiles used. The line-of-sight rangedistance and the inclination angle of a target relative to the shootermay be measured or estimated in the field where the target isencountered.

In accordance with exemplary embodiments described herein, many hunters(including bow hunters) and other shooters, such as military and lawenforcement snipers, are versed in holdover techniques for compensatingfor ballistic drop in horizontal fire scenarios. A holdover adjustmentinvolves aiming high by a measured or estimated amount. For example, acountersniper shooting a rifle with a riflescope sighted in at 200 yardsmay know that a killing shot for his target (in the heart-lung area) ata level-fire range of approximately 375 yards involves aiming the crosshairs of the riflescope at the top of the target's head. Holdoveradjustments are much faster in practice than elevation adjustments,which involve manually adjusting an elevation setting of the riflescopeor other aiming device to change the elevation angle α of the aimingdevice relative to the weapon. Holdover adjustments are also the primarymode of aiming adjustment for most archers. Holdover and holdundertechniques also avoid the need to re-zero the aiming device after makinga temporary elevation adjustment.

Many varieties of ballistic reticles are employed in riflescopes tofacilitate holdover and holdunder. For archery, a common ballisticaiming sight, known as a pin sight, is often employed for holdoveraiming adjustment. Ballistic reticles and other ballistic aiming sightsgenerally include multiple aiming marks spaced apart along a verticalaxis. Exemplary ballistic reticles include mil-dot reticles andvariations, such as the LEUPOLD TACTICAL MILLING RETICLE™ (TMR™)available from Leupold & Stevens, Inc., Leupold® DUPLEX™ reticles; theLEUPOLD SPECIAL PURPOSE RETICLE™ (SPR™); and LEUPOLD BALLISTIC AIMINGSYSTEM™ (BAS™) reticles, such as the LEUPOLD BOONE & CROCKETT BIG GAMERETICLE™ and the LEUPOLD VARMINT HUNTER'S RETICLE™ BAS reticles andmethods of using them are described in U.S. Pat. No. 7,603,804 B2 toZaderey et al., entitled “Ballistic Reticle for Projectile Weapon AimingSystems and Method of Aiming” (“the '804 patent”), the disclosure ofwhich is incorporated herein by reference. As described in the '804patent, BAS reticles include secondary aiming marks that are spaced atprogressively increasing distances below a primary aiming mark andpositioned to compensate for ballistic drop at preselected regularincremental ranges for a group of ammunition having similar ballisticcharacteristics.

In accordance with one exemplary embodiment depicted in FIGS. 2 and 3, amethod 300 of inclined shooting involves calculation of an equivalenthorizontal range (EHR) that may be used by a shooter to make a holdoveror elevation adjustment for accurately aiming a projectile weapon at anelevated or depressed target located at a inclined line-of-sight (LOS)range that is different from the EHR. With reference to FIG. 2, ashooter at vantage point VP determines a line-of-sight range to atarget. As in FIG. 1, a zero range R₀ represents the horizontal-firedistance at which the trajectory of the projectile launched from theprojectile weapon and the line of sight from the aiming device of theweapon intersect. Line-of-sight ranges R₁ and R₂ to two differenttargets are depicted in FIG. 2, illustrating the usefulness of themethod with respect to both positive and negative ballistic path heightsBP₁ and BP₂ relative to the inclined-fire LOS. For purposes ofillustration, the steps of method 300 (FIG. 3) will be described withreference to a generic LOS range R to a target T, shown in FIG. 2 atrange R₂. It should be appreciated that the methods described herein areequally applicable to “near” LOS ranges R₁ at which the ballistic pathheight BP₁ is positive, as well as to “far” LOS ranges R₂ at which theballistic path height BP₂ is negative. The LOS range R may be determinedby a relatively accurate ranging technique, such as use of a lidar(laser ranging) or radar, or by a method of range estimation, such asoptical range estimating methods in which a distant target of known sizeis bracketed in a scale of an optical device, as described in the '804patent.

Methods 300 in accordance with the present disclosure also involvedetermining an inclination θ of the inclined LOS between vantage pointVP and the target F. The angle of inclination θ may be determined by anelectronic inclinometer, calibrated tilt sensor circuit, or othersimilar device. For accuracy, ease of use, and speed, an electronicinclinometer for determining the angle of inclination θ may be mountedin a common housing with a handheld laser rangefinder 800 of the kinddescribed below with reference to FIGS. 8-10.

FIG. 3 is a flow diagram depicting steps of inclined shooting method300, including the initial steps of determining the LOS range R (step312) and determining the inclination θ of the inclined LOS (step 314).With reference to FIG. 3, after LOS range R and inclination θ have beendetermined (steps 312 and 314), method 300 may involve a check (step316) to determine whether the absolute value of inclination θ is lessthan a predetermined limit under which the effects of inclination can bedisregarded and the LOS range R can be regarded as the EquivalentHorizontal Range (EHR) (step 318).

Archery ballistics exhibit a more significant difference betweenpositive and negative lines of initial trajectory (uphill and downhillshots) because the initial velocity is relatively low, giving theeffects of gravity more time to affect the trajectory over a givendistance than with bullets, which reach their targets much faster thanarrows. Especially at long ranges, uphill shots experience more dropthan downhill shots; therefore, when applying method 300 for archery,check 316 may involve comparing a positive inclination θ against apositive limit and a negative inclination θ against a negative limitthat is different from the positive limit. Mathematically, such a checkwould be expressed as: {lower_limit}>θ<{upper_limit}?

If the result of check 316 is negative, then a predicted trajectoryparameter TP is calculated or otherwise determined at the LOS range fora preselected projectile P shot from vantage point VP toward the targetT (step 320). Trajectory parameter TP may comprise any of a variety oftrajectory characteristics or other characteristics of a projectile thatare calculable using ballistics software. For example, trajectoryparameter TP at LOS range R may comprise one or more of ballistic pathheight (e.g., arrow path or bullet path), ballistic drop relative toline of initial trajectory (e.g., the bore line in FIG. 1), observedballistic drop perpendicular to LOS (i.e., (vertical ballistic drop)cos(θ+α)), velocity, energy, and momentum. In accordance with theexemplary embodiment described below with reference to FIGS. 2 and 4,for R=R₂, trajectory parameter TP may comprise ballistic path height BP₂(e.g., bullet path height). In another embodiment, described below withreference to FIG. 5, the trajectory parameter of ballistic path heightcomprises arrow path height (AP).

Nothing in the figures or written description should, however, beconstrued as limiting the scope of possible trajectory parameters toonly ballistic path height. Of the many possible choices of trajectoryparameters, ballistic path height (bullet path height or arrow pathheight) usually is best for the shooter, who needs to know where theprojectile will be relative to the line of sight when the projectilereaches the line-of-sight range distance to the target so that theshooter can make appropriate aiming adjustments.

After the trajectory parameter TP has been calculated, the method maythen output the trajectory parameter TP (step 321) or calculate EHRbased on the trajectory parameter TP or parameters (step 322). At step321, the trajectory parameter TP output may comprise ballistic pathheight BP expressed as a linear distance in inches or millimeters (mm)of apparent drop, or as a corresponding angle subtended by the ballisticpath height (e.g., BP₂ in FIG. 2) in minutes of angle (MOA) ormilliradians (mils). The TP output (step 321) may comprise a display ofnumerical ballistic path data in an electronic display device, such as adisplay 900 (FIG. 9) of rangefinder 800 (FIG. 8) or a reticle in ariflescope. Alternatively or additionally, the TP output (step 321) maycomprise a graphical display of a holdover aiming recommendation in arangefinder display, a riflescope reticle, an archery sight, or anotheraiming sight, based on the trajectory parameter of ballistic path heightBP.

In one exemplary method of calculating EHR, a reference ballisticsequation for a level-fire scenario (θ=0) comprising a polynomial seriesis reverted (i.e., through series reversion) to solve for EHR based on apreviously calculated ballistic path height BP (BP₂). As depicted inFIG. 2, BP₂ corresponds to EHR₂ under level-fire conditions. Thus, EHRis calculated as the range at which trajectory parameter TP would occurif shooting projectile P in a level-fire condition from the vantagepoint VP toward a theoretical target T_(th) in a common horizontal planewith vantage point VP, such that the horizontal plane coincides with thelevel-fire LOS. The reference ballistics equation may be established todeviate slightly from horizontal without appreciable error.Consequently, the terms “horizontal,” “level-fire LOS,” and othersimilar terms are construed to allow for equations to deviate fromperfect horizontal unless the context indicates otherwise. For example,when solving for EHR, the degree of levelness of the reference equationsshould facilitate calculation of EHR with sufficient accuracy to allowaiming adjustments for inclined shooting resulting in better than ±6inches of error at 500 yards throughout the range of between −60 and +60degrees inclination. Ballistic trajectories are generally flatter atsteeper shooting angles and trajectories of different projectiles aretherefore more similar. Consequently, the deviation tends to be lesssignificant at very steep inclines.

The calculation of trajectory parameter TP, the calculation ofequivalent horizontal range EHR, or both, may also be based on aballistic coefficient of the projectile P and one or more shootingconditions. The ballistic coefficient and shooting conditions may bespecified by a user or automatically determined at step 324.Automatically-determined shooting conditions may include meteorologicalconditions, such as temperature, relative humidity, and/or barometricpressure, which may be measured by micro-sensors in communication with acomputer processor for operating method 300. Meteorological conditionsmay also be determined by receiving local weather data via radiotransmission signal, received by an antenna and receiver in associationwith the computer processor. Similarly, geospatial shooting conditions,such as the compass heading of the LOS to the target and the geographiclocation of the vantage point VP (including latitude, longitude,altitude, or all three), may be determined automatically by a GPSreceiver and an electronic compass sensor in communication with thecomputer processor, to ballistically compensate for the Coriolis effect(caused by the rotation of the Earth). Alternatively, suchmeteorological and geospatial shooting conditions may be specified by auser and input into a memory associated with the computer processor,based on observations made by the user. It may be noted that for bows,geospatial conditions are unnecessary because maximum range distancesare short, while for high-powered rifles, Coriolis corrections to atrajectory are necessary only if range distances exceed about 1000 yardsor meters.

User selection of shooting conditions and ballistic coefficient may alsoinvolve preselecting or otherwise inputting non-meteorological andnon-geospatial conditions for storage in a memory associated with acomputer processor on which method 300 is executed. The ballisticcoefficient and certain shooting conditions, such as the initialvelocity of projectile P (e.g., muzzle velocity, in the case ofbullets), may be set by a user simply by selecting from two or moreweapon types (such as guns and bows), and from two or more ballisticgroupings and possibly three, four, five, six, seven or more groups,such that each group has a nominal ballistic characteristicrepresentative of different sets of projectiles having similar ballisticproperties. The sets groups) may be mutually-exclusive or overlapping(intersecting). A sighted-in range of a weapon aiming device and aheight of the weapon aiming device above a bore line of a weapon mayalso be entered in this manner. In a rangefinder device 800 foroperating the method, described below with reference to FIGS. 8 and 9,the weapon type and ballistic group may be selected from a menu ofpossible choices during a menu mode or setup mode of rangefinder device800.

After a trajectory parameter TP has been calculated at step 320 or EHRhas been calculated at step 322, method 300 then involves outputting TPor EHR in some form (step 321 or 326). For example, TP or EHR may bedisplayed via a display device, such as an LCD display, in the form of anumeric value specified in a convenient unit of measure. For example, TPoutput may be expressed as ballistic path height BP in inches or mm ofapparent drop or as an angle (in MOA or mils) subtended by the ballisticpath height BP. EHR may be expressed in yards or meters, for example. Inother embodiments, BP or EHR may be effectively output via a graphicalrepresentation of the data through the identification of a reticleaiming mark corresponding to the holdover or holdunder (holdover orholdunder is always the negative of the BP), or the EHR, for example, asdescribed below.

Once EHR is output at 326, EHR can then be employed to aim theprojectile weapon at target T (step 328) along the inclined LOS at R₂.In one embodiment, a shooter merely makes a holdover or holdunderadjustment based on the calculated EHR, as if he or she were shootingunder level-fire conditions—it being noted that wind effects, firearminaccuracy, and wiggle of the shooter are still in effect over theentire LOS range R₂. In another embodiment, the shooter adjusts anelevation adjustment mechanism of a riflescope or other aiming devicebased on the displayed EHR. Similar elevation adjustments may be madebased on the display of the calculated trajectory parameter TP (step321).

FIG. 4 summarizes details of one possible sequence of steps 400 forcalculating a trajectory parameter of bullet path height (BP) andequivalent horizontal range (EHR) for bullets. The calculation sequence400 begins with selection of a ballistic group (A, B, or C) in which thebullet and cartridge are listed (step 401). Ballistic grouping mayeffectively normalize groups of bullets having similar characteristics,based on their ballistic coefficients, muzzle velocities and masses.Listings of cartridges in the various groupings may be provided to theuser by a printed table or software-generated information display,facilitating selection of the appropriate ballistic group. Referencetrajectories for ballistic groups A, B, and C are set forth in Table 1.The other inputs to the calculations include the LOS range R and theinclination angle θ, which may be determined automatically by a handheldlaser rangefinder with inclinometer (step 402). The calculation methodinvolves solving the following polynomial equation for bullet pathheight BP:BP=a ₀ +a ₁ R+a ₂ R ² +a ₃ R ³ +a ₄ R ⁴  (1)

(step 406), in which the coefficients a₀, a₁, a₂, a₃ and a₄ arecalculated from the inclination angle θ based on a series of polynomialequations 404 in which the coefficients thereof (identified in FIG. 4 asA₀₀, A₀₁, A₀₂, etc.) are different stored parameters for each ballisticgroup A, B, and C. A single equation 406 (Equation (1)) is suitable forboth positive and negative angles of inclination, expressed as absoluteangular values. After bullet path height BP has been determined, the BPis then used as an input to one of two different reversions of thebullet path equation for θ=0 to solve for EHR. If bullet path height BPis positive (test 408), then a “short-range EHR” polynomial equation isused (step 410), such that B₀, B₁, . . . , B₆ are parameterscorresponding to the selected ballistic group. If BP is negative (test408), then a “long-range EFIR” polynomial equation is used (step 412),such that C₀, C₁, . . . , C₆ are parameters corresponding to theselected ballistic group. Each ballistic group also has an associatedcoefficient named BPLIM, which is an upper limit for BP in thecomputations shown in FIG. 4. Parameters A₀₀ to A₄₃, B₀ to B₆, and C₀ toC₆ are constants that are stored for each of the ballistic groups andrecalled based on the selected ballistic group for purposes completingthe calculations 400.

FIG. 5 illustrates a similar sequence of calculations 500 for archery.In FIG. 5, reference numerals 501, 502, 506, etc., indicate steps thatrespectively correspond to steps 401, 402, 406, etc., of FIG. 4. Unlikethe calculations 400 (FIG. 4) for bullets, the calculation of ballisticpath for arrows 500 (hereinafter arrow path (AP) must take into accountwhether the inclination angle is positive or negative (branch 503), dueto the increased flight time of arrows and attendant increased effectsof gravity on their trajectory. For this reason, the calculationsinvolve one of two different sets of coefficients A_(ij) and D_(ij),(for 3, 4, 5 and j=1, 2, 3, 4, 5) depending on whether the inclinationis positive ((step 504 a′) or negative (step 504 b′). Parameters A₀₀ toA₄₃, B₀ to B₆, C₀ to C₆, D₀₀ to D₄₃, APLIM, and EHRLIM are constantsthat are stored in memory for each of the ballistic groups and recalledbased on the selected ballistic group for purposes completing thecalculations 500.

Table 1 lists one example of criteria for ballistic grouping of arrowsand bullets:

TABLE 1 BALLISTIC CHARACTERISTIC GROUP BALLISTIC DROP (WITHOUT INCLINE)Arrow group A Arrow drop of 20-30 in from the 20 yd sight pin at 40 ydArrow group B Arrow drop of 30-40 in from the 20 yd sight pin at 40 ydArrow group C Arrow drop of 10-20 in from the 20-yd sight pin at 40 ydBullet group A Rifles sighted in at 200 yards with 30-40 in drop at 500yd Bullet group B Rifles sighted in at 200 yards with 40-50 in drop at500 yd Bullet group C Rifles sighted in at 300 yards with 20-30 in dropat 500 yd

Arrow groupings may be more dependent on the launch velocity achievedthan the actual arrow used, whereas bullet groupings may be primarilybased on the type of cartridge and load used. Table 2 lists exemplaryreference trajectories from which the calculation coefficients of FIG. 4may be determined for ballistic groups A, B, and C.

TABLE 2 BALLISTIC GROUP REFERENCE TRAJECTORY A Winchester Short Magnumwith Winchester 180 grain Ballistic Silvertip bullet at 3010 fps, havinga level fire bullet path height of −25.21 in at 500 yds. B 7 mmRemington Magnum with Federal 150 grain SBT GameKing bullet at 3110 fps,having a level fire Bullet Path height of −34.82 in at 500 yds. C 7mm-08 Remington with Remington Pointed Soft Point Core-Lokt bullet at2890 fps, having a level fire Bullet Path height of −45.22 in at 500yds.

Alternatives to solving a series of polynomial equations also exist,although many of them will not provide the same accuracy as solving apolynomial series. For example, a single simplified equation forballistic drop or ballistic path height may be used to calculate apredicted trajectory parameter, and then a second simplified equationused to calculate EHR from the predicted trajectory parameter. Anotheralternative method of calculating EHR involves the “Sierra Approach”described in W. T. McDonald, “inclined Fire” (June 2003), incorporatedherein by reference. Still another alternative technique for calculatingballistic drop or ballistic path involves a table lookup of a predictedtrajectory parameter and/or interpolation of table lookup results,followed by calculation of EHR using the formula identified in FIG. 4.Yet another alternative involves determining both the predictedtrajectory parameter and EHR by table lookup and interpolation, usingstored sets of inclined-shooting data at various angles.

Table 3 illustrates an example of an EHR calculation using the sequenceof steps for calculating a trajectory parameter of bullet path height(BP) and equivalent horizontal range (EHR) for bullets described abovein connection with FIG. 4. The EHR calculations in Table 3 are alsocompared with the results of aiming using EHR to aiming with nocompensation for incline, and aiming by utilizing the horizontaldistance to the target (rifleman's rule).

TABLE 3 .300 WSM, 165 GRAIN NOSLER PARTITION, 3050 FPS MUZZLE LOADVELOCITY Angle of inclination 50° Inclined line-of-sight range 500 ydsEquivalent Horizontal Range (EHR) 389 yds Ballistic table holdover for389 yds 18 in level fire Horizontal leg of the triangle 321 ydsBallistic table holdover for 321 yds 8.5 in Error if horizontal leg isused −9.5 in Ballistic table holdover for 500 yds 39.5 in level fire (nocompensation for incline) Error if no compensation for incline +21.5 ins

The subject matter disclosed herein also provides a technique fordetermining highly accurate trajectories for inclined shooting thataccounts for effects to a trajectory caused by altitude. That is, thesubject matter disclosed herein also provides a technique fordetermining highly accurate Altitude-Compensated Inclined Shooting(ACIS) trajectories. According to the subject matter disclosed herein,ACIS trajectories are determined for a specific cartridge type at alltarget range distances within the maximum effective range of thecartridge, at all positive and negative target inclination angles within±70 degrees, and at all altitudes above sea level up to a practicalmaximum for first zeroing-in the firearm and then later firing (at adifferent altitude) at a target encountered in the field. Additionally,the ACIS trajectories provide a full 3-degree-of-freedom analyticalmodel for trajectory calculations. Moreover, the firearm may be amachine gun, rifle, or handgun using that specific cartridge type. Theexemplary ACIS method described herein is described for firearms thatshoot bullets. The ACIS method also may be applied to bows shootingarrows, although some modifications are necessary because, as explainedabove, an arrow trajectory for shooting upward at a positive inclinationangle differs from the trajectory for shooting downward at a negativeinclination angle of the same value, which is caused by lower velocitiesand longer times of flight for arrows over a given distance,

The computational procedure for ACIS trajectories utilizescharacteristics of known firearm and ammunition, such as projectileballistic coefficient(s) available from manufacturers' data or othertest data, and the muzzle velocity of the firearm. Additionally, thecomputational procedure utilizes standard meteorological conditions,i.e., standard atmospheric pressure, temperature, and relative humidityversus altitude, which are available from the Army StandardMeteorological Atmosphere (“Army Standard Metro”) at the particularaltitude at which the firearm is sighted-in (zeroed-in) and at theparticular altitude of the firing point in the field. For Army StandardMetro information, see, for example, “Modern Exterior Ballistics,” R. L.McCoy, Schiffer Military History, 1999, page 166; “Exterior Ballisticsof Small Arms Projectiles,” E. D. Lowry, Olin Mathiesin ChemicalCorporation, 1965, page 74; and “Sierra Rifle Reloading Manual,” 3^(rd)Edition, Sierra Bullets L. P., 1989, pages 480-481). The Army StandardMetro atmosphere is the reference for the standard drag functions G1,G5, G6, G7, etc., and the ballistic coefficient value(s) pertaining toeach of those drag functions and, consequently, is the referenceatmosphere for almost all predictive ballistic computations. Otherpossibilities exist for atmospheric conditions, such as theInternational Standard Atmosphere (ISA) or even absolute measurements oftemperature, pressure, and humidity at the shooting location, but theArmy Standard Metro Atmosphere has been adopted for predictive ballisticcomputations with commercial ammunition. Altitude information above sealevel at the zeroing-in site and at the firing site is obtained fromreference data, field measurements, topographical maps, etc. Targetconditions (i.e., target direct-range distance and target inclinationangle relative to the firing point) are obtained from measurements madein real time when a target is encountered in the field. Winds are notconsidered because wind conditions, particularly at the firing location,cannot be accurately predicted in advance. Consequently, when a shooteris using trajectory information provided by the computational proceduredisclosed herein, the experience of the shooter must be relied on tocorrect for wind conditions when a target is encountered in the field.

In one exemplary, embodiment, the computational technique disclosedherein utilizes two different computer processors, such as a relativelyhigh-numerical-precision computer processor (referred to herein as a“master processor”) and a relatively low-numerical-precision computerprocessor (referred to herein as a “device processor”). The masterprocessor is used to pre-compute reference trajectory information thatis later used by the device processor to provide near real-time (1second or less) Altitude-Compensated Inclined Shooting (ACAS) trajectoryinformation in the field. Exemplary embodiments of the master processorcomprise, but are not limited to, a Personal Computer (PC) or a handheldPersonal Digital Assistant (PDA) having ballistics software capable ofcomputing a highly accurate projectile trajectory. Exemplary embodimentsof a device processor comprise, but are not limited to, a computerprocessor device having a limited computation capacity, such as a devicemounted to a firearm or a handheld device used by a shooter or acompanion of the shooter. In an alternative exemplary embodiment, asingle computer processor comprising sufficient computing capability maybe used in place of two different computer processors. An exemplaryembodiment of such a single computer processor comprises, but is notlimited to, a handheld Personal Digital Assistant (PDA) havingballistics software capable of computing a highly accurate projectiletrajectory. It should understood that the computer processors referredto herein generally include components and capability for providingfunctionality, such as, but not limited to, input/output (I/O), storage,power, etc.

Because the trajectory information provided by the computationalprocedure applies to a single cartridge, the memory storage requirementof the device processor for each cartridge is relatively modest becauseonly reference trajectory information for the particular cartridge needsto be stored for the device processor. If reference trajectoryinformation for multiple cartridges is desired, the reference trajectoryinformation for each desired cartridge is generated by the masterprocessor by repeating the computational procedure for each desiredcartridge and then transferring the different reference trajectoryinformation for each cartridge to the device processor. Memoryrequirements for the device processor would accordingly increase basedon the number of desired cartridges. In the field, the referencetrajectory information is then accessed by an operator through thedevice processor. The computational procedure performed by the deviceprocessor for determining an ACIS trajectory is the same regardlesswhich cartridge is being used.

In an alternative exemplary embodiment, all computations are performedby a master-processor-type computer processor, such as a handheldPersonal Digital Assistant (PDA) having ballistics software capable ofcomputing a highly accurate projectile trajectory. For this alternativeexemplary embodiment, reference trajectory information is computed priorto firing in the field, and then accessed in the field for computing anACIS trajectory for a selected cartridge.

FIG. 6 depicts a flow diagram 600 for one exemplary embodiment ofoperations and a computational process performed by a master processorfor generating reference trajectory information for computingAltitude-Compensated Inclined Shooting (ACS) trajectory information fora selected cartridge. Computations are initiated at 601 by, for example,a user inputting and/or selecting from a menu the initial datacomprising (1) the specific projectile for which the computations willbe performed; (2) the ballistic coefficient(s) for the projectile andthe projectile speed range within which each ballistic coefficient valueapplies; (3) the muzzle velocity (i.e., the speed of the projectile whenthe projectile leaves the muzzle of the firearm); (4) the zero-rangedistance for which the firearm has been or will be sighted-in; (5) themaximum range distance for which trajectory computations are to beperformed (normally the maximum effective range distance for thecartridge); (6) the range-distance increment (RDI) (e.g., 50 yards ormeters for a high powered cartridge) at which trajectory parameters willbe outputted, listed, and/or stored for the projectile; (7) the sightheight of the firearm (i.e., the perpendicular distance of the line ofsight through the sighting device on the firearm above the borecenterline; and (8) the physical units in which trajectory parameterswill be expressed (i.e., all English units, all metric units, or “mixed”units in which range distances are expressed in meters and all otherparameters are expressed in English units). The following descriptionuses English units, but it should be understood that metric or mixedunits could also be used.

At 602, the Army Standard Meteorological conditions are selected for allcomputations that will be performed. In one exemplary embodiment, theArmy Standard Metro conditions are preloaded into the master processorbefore computations can begin. In another exemplary embodiment actualatmospheric conditions can be selected and, for example, manuallyentered when atmospheric conditions can be practically predicted inadvance of going into the field. At sea level, the meteorologicalconditions are 750 mm (29.5275 in) of mercury atmospheric pressure, 15C(59F) atmospheric temperature, and 78% relative humidity. The pressureand temperature at higher altitudes decrease in accordance with tableslisted in the publications referred to previously. The value of relativehumidity normally is not changed as altitude changes because relativehumidity has a small effect on air density at sea level, and as altitudeincreases, the vapor pressure of water in the atmosphere decreases,leading to an even smaller effect caused by relative humidity. Astandard value of gravitational acceleration of 32.174 ft/sec² adjustedfor altitude at the firing point is also used in all computations.

At 603, a baseline trajectory is computed for the selected projectile.The baseline trajectory is a level-fire (zero-inclination angle)trajectory at sea-level standard conditions between the muzzle of thefirearm and the maximum-range distance (entered at 601). The parameterof interest is the Bullet Path Height (BPH) of the projectile versusrange distance from the muzzle. BPH is defined herein as theperpendicular distance of the projectile from the extended line of sightthrough the sighting device on the firearm. It should be understood thatBPH is not the Drop of the projectile. For a level-fire trajectory, Dropis defined herein as the distance of the bullet from a level linebetween the muzzle of the firearm and a target located in a level planewith the firearm. As defined herein, BPH is positive when the projectileis above the line of sight and negative when the projectile is below theline of sight. The positive or negative sign of BPH allows an operatorto know where the projectile will pass with respect to the extended lineof sight through the sighting device on the firearm.

Both BPH and Drop are routinely calculated by software that computestrajectories for projectiles in fast and high numerical-precisioncomputers, such computer processors like the master processor. Anexample of such software is the Sierra Infinity software, which isavailable from Sierra Bullets, Sedalia, Mo. For a level-fire trajectory,BPH and Drop at any given range distance are related by an algebraicequation so that only one variable need be known for the purposes of thecomputational process disclosed herein. For illustrative purposes, BPHis used in the following description. The zero-range distance R_(Z)selected for the firearm and the Drop D_(Z), at that specific rangedistance are stored in the master processor for later transfer to thedevice processor.

At 604, having BPH versus range distance at selected ranges (e.g., every50 yards) between the muzzle of the firearm and the maximum-rangedistance, the master computer fits (either internally or by using aseparate external software program) a polynomial expression to the BPHversus range distance, such as by using the least squares method of fit.According to one exemplary embodiment, the standard deviation of the fitshould be no greater than 0.5 inch. In one exemplary embodiment, aseventh-order polynomial is sufficient for most projectile trajectories.If the trajectory is very flat, a lower-order polynomial could besufficient. For trajectories that are more steeply curved, especiallynear the end of the trajectory, or to reduce the number of terms in thefit polynomials to simplify the computations, sequential sectors of thetrajectory may be specified, and a different fit polynomial may be usedin each sector. In one exemplary embodiment, no more than three sectorsare necessary, but in alternative exemplary embodiments, a greater orlesser number of sectors could be used without restriction. For aseventh-order polynomial, the reference Bullet Path (BP) polynomial forany sector will be:BP(R)=a _(i0) +a _(i1) R+a _(i2) R ² +a _(i3) R ³ +a _(i4) R ⁴ +a _(i5)R ⁵ +a _(i6) R ⁶ +a _(i7) R ⁷  (2)in which, i is an index indicating the trajectory segment of the fit(i=1, 2, or 3); and R is the range distance.

In one exemplary embodiment, the summation is over all eight terms inthe polynomial expansion. In an alternative exemplary embodiment, thesummation is over fewer terms of the polynomial expansion. Thepolynomial fit operation yields the eight (or fewer) coefficients a_(ij)in which j=0, 1, 2, . . . , 7) of the polynomial fit in each sector ofthe baseline trajectory. The coefficients a_(ij) are stored forsubsequent transfer to the device processor. In the explanation thatfollows, the seventh-order polynomial of Equation (2) will be used.

At 605, the next group of computations performed by the master processortakes place at a specific set of range distances from the firearm, thatis, at each range-distance increment (RDI) between the maximum rangedistance of the trajectory back to a range distance that is onerange-distance increment RDI beyond the zero-range distance R_(Z) of thefirearm. For example, if the maximum range distance is 800 yards, theR_(Z) is 200 yards, and the RDI is 50 yards, trajectory calculationstake place at range distances of 800, 750, 700, 650, 600, . . . , downto 250 yards, resulting in calculations at twelve specific rangedistances. The specific range distances are designated by R_(k) in thefollowing explanation. At each such specific range distance, the masterprocessor uses the specified zero-range distance R_(Z) with noadjustment in sight settings to compute the BPH for shooting at sealevel altitude, at 2000 feet above sea level (ASL), at 4000 feet ASL, at6000 feet ASL, at 8000 feet ASL, at 10000 ASL, and at 12000 feet ASL.The trajectory calculated for shooting at sea level is used as thebaseline reference trajectory. This observation is crucial; the shooteralways uses a specific value of zero-range distance R_(Z) (e.g., 200yards) regardless of the altitude at which he or she zeroes-in thefirearm.

The maximum shooting altitude anticipated for zeroing-in and/or shootingin the field is selected for the present explanation to be 12000 feetASL. It should be understood that in an alternative exemplaryembodiment, the maximum shooting altitude could be selected to bedifferent from 12000 feet ASL, in which case the maximum altitude wouldbe appropriately adjusted for the anticipated shooting situation in thefield. Normally, for practical reasons one exemplary embodiment of thesoftware in the master processor will be limited to a maximum altitudeof 15000 feet ASL. For the present exemplary embodiment, an altitudeseparation of 2000 feet is used. It should be understood that in analternative exemplary embodiment, a different altitude separation couldbe used, such as a value that is less than 2000 feet, in yet anotherexemplary embodiment, the attitude separation value could vary, that is,not be a constant value.

The computations at 605 result in a set of seven (or fewer depending onthe maximum shooting altitude and the altitude separation value) BulletPath (BP) values at each range distance from one range-distanceincrement (RDI) beyond R_(Z) out to the maximum range specified for theprojectile. Note that no polynomial fits are required for thetrajectories computed at 605. The results of the computations are a setof seven (or fewer) Bullet Path (BP) values at each specific rangedistance R_(k) chosen for the evaluation. Each such computation is basedon zeroing-in (sighting-in) the firearm at sea level and then shootingat each altitude of the list without any sight changes to compensate forthe altitude changes.

At 606, the master processor calculates a set of BPH Corrections at eachspecific range distance R_(k) and each altitude. This is done bydesignating the BPH for each range distance R_(k), at sea level and ateach altitude above sea level (ASL) as:BP₀(R _(k))=BPH at R _(k) fired at sea level;BP₂₀₀₀(R _(k))=BPH at R _(k) fired at 2000 feet ASL;BP₄₀₀₀(R _(k))=BPH at R _(k) fired at 4000 feet ASL;BP₆₀₀₀(R _(k))=BPH at R _(k) fired at 6000 feet ASL;BP₈₀₀₀(R _(k))=BPH at R _(k) fired at 8000 feet ASL;BP₁₀₀₀₀(R _(k))=BPH at R _(k) fired at 10000 feet ASL; andBP₁₂₀₀₀(R _(k))=BPH at R _(k) fired at 12000 feet ASL.  (3)

The left side of Equations (3) are the Ballistic Path Heights fortrajectories in which the firearm is sighted-in at sea level, then laterfired at the specified altitudes and evaluated at the range distanceR_(k). The Bullet Path Height Correction BPcorr(R_(k)) at range distanceR_(k) and at each altitude is given by the following arithmeticoperations of Equation (4):BPcorr₀(R _(k))=0;BPcorr₂₀₀₀(R _(k))=BP₂₀₀₀(R _(k))−BP₀(R _(k));BPcorr₄₀₀₀(R _(k))=BP₄₀₀₀(R_(k))−BP₀(R _(k));BPcorr₆₀₀₀(R _(k))=BP₆₀₀₀(R _(k))−BP₀(R _(k));BPcorr₈₀₀₀(R _(k))=BP₈₀₀₀(R _(k))−BP₀(R _(k));BPcorr₁₀₀₀₀(R _(k))=BP₁₀₀₀₀(R _(k))−BP₀(R _(k));BPcorr₁₂₀₀₀(R _(k))=BP₁₂₀₀₀(R _(k))−BP₀(R _(k));  (4)

As shown by Equation (4), the Bullet Path Height Correction BPcorr ateach firing altitude and at the specific range distance R_(k) is thedifference between the BPH at which the projectile is fired and the BPHof the projectile when sighted-in at sea level. The Bullet Path HeightCorrection BPcorr is applied to the Bullet Path Height BPH of thereference trajectory at the range distance R_(k). If the projectile wereto be fired at sea level, the correction for altitude would be zero atall range distances because the reference trajectory is the actualtrajectory at that (zero) altitude.

A set of specific range distances R₁, R₂, R₃, . . . , R_(K) is chosen sothat each range point R_(k) corresponds to one of the output points forthe computations of Equation (2). ThenR ₁ =R _(Z)+RDIR ₂ =R ₁+RDIR ₃ =R ₂+RDI. . .R _(K) =R _(K-1)+RDI  (5)in which, R_(K) equals the maximum range distance for the trajectorycomputation.

The sequence starts one range-distance increment (RDI) beyond thezero-range distance R_(Z) because the projectile trajectory rises only alittle above the line of sight at points between the muzzle and thezero-range distance R_(Z). This is because the zero-range distance R_(Z)for the firearm is chosen so that a target miss will not occur for adirect aim at a target closer than R_(Z). For an “inclined target”elevated or depressed relative to the firing point, the projectiletrajectory is flatter than it is for zero inclination, and no BPcorr isneeded for target distances less than R_(Z).

The computations of the equations of Equation (5) result in a list ofBPcorr values at each specified range distance R_(k) and at eachaltitude chosen for evaluation. The actual number of such lists for eachaltitude is given by:

$\begin{matrix}{K = \frac{R_{k} - R_{Z}}{RDI}} & (6)\end{matrix}$

At this point in the computations, a second-order, or at most athird-order, polynomial is fitted to the BPcorr values versus altitudein the list above resulting in a polynomial of the form:BPcorr(R _(k),alt)=c _(1k)(R _(k))x(alt)+c _(2k)(R _(k))x(alt)² +c_(3k)(R _(k))x(alt)³  (7)

Equation (7) expresses the Bullet Path Height Correction BPcorr at aspecific range distance R_(k) and any altitude (alt) as a power seriesin which the coefficients c_(nk)(R_(k)) change values at each specificrange distance R_(k). The initial value c_(0k)(R_(k)) usually appearingin the polynomial of Equation (7) is always 0 because the Bullet PathHeight Corrections for zero altitude are zero at all range distances. Ifonly a second-order polynomial is necessary (which is usually the case),then the third term on the right side of Equation (7) does not appear.There would then be K such polynomials for a given projectile. For thepresent example in which the maximum-range distance R_(K) is 800 yards,the zero range distance R_(Z) is 200 yards, and the range-distanceincrement RDI is 50 yards, there would be twelve polynomials of the formof Equation (7) to characterize the BPcorr at 800, 750, 700, 650, . . ., down to 250 yards (or meters): That is,

$\begin{matrix}{K = {\frac{800 - 200}{50} = 12.}} & (8)\end{matrix}$

At this point, the calculations performed by the master processor havebeen completed. The coefficients a_(ij) in Equation (2) and the K setsof coefficients c₁(R), c₂(R), and c₃(R) for Equation (7) are stored forlater transfer to the device processor at 607 at a convenient time.

Few shooters sight in exactly at sea level. A shooter normally sights-ina firearm at a convenient location at some altitude, referred to hereinas the zero-point altitude (alt_(ZP)), which is almost always above sealevel. Then, in the field the shooter fires at a target at yet adifferent altitude, referred to herein as firing-point altitude(alt_(FP)) if the shooter has a reference trajectory computed for sealevel, two Bullet Path Height corrections BPcorr are necessary to modifythe reference trajectory to predict the projectile impact point at thefield location:

The first correction modifies the sea level reference trajectory to thealtitude alt_(ZP) at which the shooter sighted in. The second correctionmodifies the reference trajectory for the altitude alt_(FP) at which theshooter must fire. It has been determined that the net Bullet PathHeight Correction BPcorr for any two altitudes and any range distance tothe target can be computed from:BPcorr(net)=BPcorr(R _(T),alt_(FP))−BPcorr(R_(T),alt_(ZP))  (9)in which, R_(T) is the target range distance with respect to the firingpoint;

-   -   BPcorr(R_(T),alt_(ZP)) is the Bullet Path Height correction        BPcorr evaluated at alt_(ZP), the sighting-in location altitude        and target range distance R_(T), evaluated by Equation (7)        above; and    -   BPcorr(R_(T), alt_(FP)) is the Bullet Path Height correction        BPcorr evaluated at alt_(FP), the firing point altitude, and the        true target range distance R_(T) evaluated by Equation (7)        above.

This crucial Equation (9) reflects an observation that the net BulletPath Height Correction for zeroing-in at a first altitude and thenfiring at a second altitude is the negative of the correction obtainedfor zeroing-in at the second altitude and then firing at the firstaltitude. Thus, Equation (9) is correct for all altitude situations.

The net Bullet Path Height Correction BPcorr(net) of Equation (9) isapplied to the original reference projectile trajectory at rangedistance R_(T) to calculate the correct Bullet Path Height BPH for thetarget at the firing location.

Note that Equations (7) and (9) are evaluated for a level-firesituation, as if the target were at a distance R_(T) in the same levelplane as the muzzle of the firearm. The target may in fad be inclined atsome angle α_(T) (positive upward or negative downward) relative to thefiring point. It has been shown in “Inclined Fire,” W. T. McDonald,2003, available from sierrabullets.com, that the correct Bullet PathHeight BPH for an inclined target can be computed from the level-firetrajectory. The appropriate equation is:BPinclined(R _(T),α_(T))=BPlevel(R _(T))cos α_(T)−(1.0−cos α_(T))[h_(S)+(R _(T) /R _(Z))(D _(Z) −h _(S))]  (10)

-   in which, BPinelined(R_(T),α_(T)) is the Bullet Path Height BPH for    a target at straight line range distance R_(T) and inclination angle    α_(T) on the inclined trajectory;    -   BPlevel(R_(T)) is the Bullet Path Height BPH computed as if the        target were at the same range distance R_(T) in a level-fire        situation;    -   h_(S) is the sight height above the bore centerline (always        positive for a sighting device mounted above the bore);    -   R_(Z) is the zero range distance for the firearm; and    -   D_(Z) is the Drop for the reference trajectory at the zero range        distance R_(Z) (available from the calculation of the reference        trajectory in the master processor).

Equation (10), when processed in a device-processor-type computerprocessor, is an excellent approximation with accuracy well within 1.0minute of angle (MoA) for range distances within the maximum effectiverange of high powered cartridges. Note that R_(Z) and D_(Z) are fixedfor a given projectile and are also transferred from the masterprocessor to the device processor. With the definitions and equationsabove, the computations performed by the device processor are littlemore than arithmetic. The trigonometric cosine function must either beavailable in the device processor, or a table of cosine values must beavailable fix the device processor for angle range of 0 degrees to 70degrees.

FIG. 7 depicts a flow diagram 700 for one exemplary embodiment ofoperations and computational process performed by a device processor forgenerating Altitude-Compensated Inclined Shooting (AGES) trajectoryinformation for a selected cartridge. In one exemplary embodiment,results of the ACES trajectory information computations performed by themaster processor would have been transferred to the device processor at607 in FIG. 6. As such, the computations performed by the masterprocessor operations need not occur in real time and could occur in aremote location in advance of a user venturing into the field becausethe computations depend only on well-known characteristics of thefirearm and ammunition and standard atmospheric conditions. Thetransferred reference trajectory information comprises:

-   -   (1) the coefficients a_(ij) of the reference trajectory Bullet        Path Height BPH from Equation (1);    -   (2) the range distance boundaries of the sectors i of the        reference trajectory within which specific a_(ij) coefficients        apply;    -   (3) the list of BPcorr coefficients c_(1k), c_(2k), and c _(3k)        for each of the k range distance points R_(k);    -   (4) the zero-range distance R_(Z) from Equation (10);    -   (5) the projectile Drop at the zero-range distance D_(Z) from        Equation (10); and    -   (6) the sight height of the firearm h_(S) from Equation (10).

At 701, before venturing into the field, the field-dependent data forgenerating Altitude-Compensated Inclined Shooting (ACIS) trajectoryinformation for a selected cartridge is input into the device processor.In particular, the altitude at which the firearm was sighted-in,alt_(ZP), and the altitude of the firing point in the field, alt_(FP),are manually input the device processor. The altitude alt_(ZP) could beentered into the device processor when the firearm is sighted-in, andthen stored in, for example, long-term memory associated with the deviceprocessor. Alternatively, could be entered into the device processorwhen the results of the computations performed by the master processorinformation are transferred into the device processor. In either case,alt_(ZP) must be entered before computations can begin at the firingpoint. The altitude of the firing point alt_(FP) normally is entered atthe time of or just prior to, encounter with a target at 702. Insituations in which a shooter can predict where the target encounterwill take place, the firing point altitude alt_(FP) may be entered intothe device processor prior to venturing into the field. The subsequentcalculations will be accurate provided that the ultimate firing pointdoes not significantly depart (about ±300 feet AST) from the predictedfiring point when the target is encountered in the field.

At 703, when a target is encountered, the shooter initiates ACIStrajectory computations. The device processor communicates with sensorsmeasuring the direct (straight line) target range distance R_(T) (i.e.,from a rangefinder) at 703 a, and the target inclination angle α_(T)(i.e., from an inclinometer) at 703 b. This communication may take placeeither (1) automatically electronically, or (2) via manual input by theshooter.

At 704, the device processor computes the Bullet Path Height BPH valuesfor the target-range distance R_(T) on a level-fire trajectory using thepolynomial Equation (2) with the values of the coefficients a_(ij)transferred from the master processor at 607 in FIG. 6. The deviceprocessor first determines the trajectory sector i in which the targetlies by comparing R_(T) with the range distance boundaries transferredfrom the master processor, then selecting the set of a_(ij) coefficientsfor that sector to perform the computations.

In the situation in which the device processor has limited numericalprecision, Equation (2) can be reformulated as a “nested polynomial” forthe evaluation computations in the following form:BP(R _(T))=a _(i0) A ₁A ₁=1+(a _(i1) /a _(i0))R _(T) A ₂A ₂=1+(a _(i2) /a _(i1))R _(T) A ₃A ₃=1+(a _(i3) /a _(i2))R _(T) A ₄A ₄=1+(a _(i4) /a _(i3))R _(T) A ₅A ₅=1+(a _(i5) /a _(i4))R _(T) A ₆A ₆=1+(a _(i6) /a _(i5))R _(T) A ₇A ₇=1+(a _(i7) /a _(i6))R _(T)  (11)

The computation of BP(R_(T)) begins with the calculation of A₇, thenproceeds to A₆, then to A₅, and so forth, to the final calculation ofBP(R_(T)). Each computation includes a truncation error based on thelimited numerical precision of the device processor. Accordingly, in thecalculation sequence of Equation (11), truncation errors occursystematically from the smallest numerical result to the largest, sothat overall truncation error in BP(R_(T)) is thereby minimized. Thesequence of Equation (11) also limits the number of multiplications thatare performed because R_(T) is not raised to powers greater than 1 inany calculation.

At 705, the device processor calculates the (fully corrected) BPcorr fora target at range distance R_(T) for a level-fire trajectory by firstdetermining between which pair of specific range distances, R_(k-1) andR_(k) the value R_(T) lies by, for example, the following sequence oftests:Is R _(T) >R ₁? If no, then R _(k-1) =R _(Z), and R _(k) =R ₁. If yes,then:Is R _(T) >R ₂? If no, then R _(k-1) =R ₁, and R _(k) =R ₂. If yes,then:Is R _(T) >R ₃? If no, then R _(k-1) =R ₂, and R _(k) =R ₃. If yes,then:. . .Is R _(T) >R _(K-1)? If no, then R _(k-1) =R _(K-2), and R _(k) =R_(K-1). If yes, then:R _(k-1) =R _(K-1), and R_(k) =R _(K).  (12)

Note that the sequence of tests of Equation (12) presumes that (a) notarget-range distance is less than the zero-range distance R_(Z) (if so,the shooter will simply fire directly at the target), and (b) that notarget-range distance is greater than the maximum range distance R_(K)specified for the firearm. If necessary, tests can be incorporated inthe device processor software to assure that these two presumptions aretrue.

With the specific range distances R_(k-1) and R_(k) known between whichR_(T) lies, at 706 the device processor next computes the Bullet PathHeight Correction BPcorr(R_(k-1), alt_(ZP)) at range distance R_(k-1)and the altitude alt_(ZP) at which the firearm was sighted in. Thiscomputation uses Equation (7) and the list of the coefficients c_(1k),c_(2k), and c _(3k) transferred from the master processor. If necessaryto preserve computational precision in the device processor, Equation(7) may be reformulated as a nested polynomial using the procedureoutlined for Equation (11). Computation as a nested polynomial isdirectly followed by the computation of the Bullet Path HeightCorrection BPcorr(R_(k),alt_(ZP)) at the specific range distance R_(k)and the altitude alt_(ZP) at which the firearm was sighted in. Then,because R_(T) lies between R_(k-1) and R_(k), the BPcorr(R_(k),alt_(ZP)) is computed by linear interpolation:

$\begin{matrix}{{{BPcorr}\left( {R_{T},{alt}_{ZP}} \right)} = {{{BPcoor}\left( {R_{k - 1},{alt}_{ZP}} \right)} + {\frac{R_{T} - R_{k - 1}}{R_{k} - R_{k - 1}}\left\lbrack {{{BPcorr}\left( {R_{k},{alt}_{ZP}} \right)} - {{BPcorr}\left( {R_{k - 1},{alt}_{ZP}} \right)}} \right\rbrack}}} & (13)\end{matrix}$

The device processor also computes at 705 BPcorr(R_(k),alt_(FP)) at thetarget range distance R_(T) and the firing point altitude alt_(FP) usingthe same procedure as for Equation (13), thereby yielding:

$\begin{matrix}{{{BPcorr}\left( {R_{T},{alt}_{FP}} \right)} = {{{BPcorr}\left( {R_{k - 1},{alt}_{FP}} \right)} + {\frac{R_{T} - R_{k - 1}}{R_{k} - R_{k - 1}}\left\lbrack {{{BPcorr}\left( {R_{k},{alt}_{FP}} \right)} - {{BPcorr}\left( {R_{k - 1},{alt}_{FP}} \right)}} \right\rbrack}}} & (14)\end{matrix}$

At 706, the net Bullet Path Height Correction is calculated usingEquation (9) above:BPcorr(net)=BPcorr(R _(T),alt_(FP))−BPcorr(R _(T),alt_(ZP))  (15)

At this point, the result of Equation (15) is the net Bullet Path HeightCorrection for a firearm and cartridge zeroed-in at an altitude alt_(ZP)and then fired at an altitude alt_(FP) at a fictitious or theoreticaltarget situated at a range distance R_(T) in a plane level with thefiring point. At 707, the net correction BPcorr(net) is added to theBullet Path Height BP(R_(T)) computed earlier for the target; rangedistance R from the range-distance sensor shown in FIG. 7. That is:BPlevel(R _(T))=BP(R _(T))+BPcorr(net)  (16)At 708, the final computation in the device processor adjusts the resultof Equation (16) for an inclined target at the same distance R_(T) andinclination angle α_(T) relative to the firing point. The final resultis:

$\begin{matrix}{{{BPinclined}\left( {R_{T},\alpha_{T}} \right)} = {{{{BPlevel}\left( R_{T} \right)}x\;\cos\;\alpha_{T}} - {\left( {1 - {\cos\;\alpha_{T}}} \right)\left\lbrack {h_{s}\; + {\frac{R_{T}}{R_{Z}}\left( {D_{Z} - h_{s}} \right)}} \right\rbrack}}} & (17)\end{matrix}$

It remains only for the device processor to display the ACIS holdoverinformation to the shooter at 709. The holdover will be the negative ofthe adjusted Bullet Path Height BPH for the inclined target because,because the projectile will be below the line of sight at rangedistances beyond the zero-range distance R_(Z), the shooter must aimhigh by an equivalent deflection at the target. Elevation adjustmentsmay be made by the shooter based on the displayed ACIS information.

A skilled mathematician will note that alternative mathematicalimplementations may be used to accomplish the detailed computationsdescribed above. All such alternatives are included herein. It isimperative, however, that the two crucial characteristics of thiscomputation method be preserved by whatever mathematical implementationis used. The first characteristic is that the same zeroing-in rangedistance R_(Z) is used at all altitudes. This is a practical convenienceto the shooter as well as essential to the mathematics; the shooter neednot attempt to adjust the zeroing-in operations for the altitude of thezeroing-in location. The second characteristic is the use of Equation(9) to compute the net Bullet Path Height Correction IsTcorr to add tothe Bullet Path Height value on the reference trajectory at target rangedistance R_(T).

The above-described methods may be implemented in a portable, handheldlaser rangefinder 800, an exemplary embodiment of which is depicted inFIG. 8. The exemplary embodiment of rangefinder 800 includes a laserranging system 804 having a lens 806 through which a laser beam isemitted and reflected laser light is received for determining a range toa distant target. Rangefinder 800 also includes an integrated opticaltargeting sight 810 comprising an objective 812 and an eyepiece 814,through which a user views the target. In one exemplary embodiment,rangefinder 800 comprises a power button 816 that turns on certainelectronics of rangefinder 800 and causes rangefinder 800 to emit laserpulses and acquire range readings. A pair of menu interface buttons 818may be provided on rangefinder 800 for operating menus for inputtingsetup information and enabling functions of the rangefinder, similar tothat described in U.S. Patent Application Publication No. 2007/0097351A1 to York et al., the disclosure of which is incorporated herein byreference. It should be understood that other alternative exemplaryembodiments of a portable handheld laser rangefinder are possible.

FIG. 9 depict one view of elements of an exemplary display 900 which, inone exemplary embodiment of rangefinder 800, may be placed in the fieldof view of the targeting sight 810 of rangefinder 800. In one exemplaryembodiment, display 900 comprises a transmissive LCD display panelplaced between objective 812 and eyepiece 814. Other display devices,however, may be used, including displays generated outside of theoptical path of the targeting sight 810 and injected into the opticalpath of the targeting sight 810, such as by projecting a reticle displayonto a prism or beam-combining element (reverse beam splitter). In oneexemplary embodiment, display 900 may comprise a circular menu 904 alongits perimeter, which can be navigated using buttons 816, 818 to selectone or more of various functions of rangefinder 800. The exemplary iconslabeled >150, 1st TGT, LAST TGT, M/FT/YD, LOS relate to rangingfunctions and/or modes of display. In one exemplary embodiment, theexemplary TBR icon may stand for TRUE BALLISTIC RANGE and, whenselected, activates calculation methods for determining ACIS holdover orthe Equivalent Horizontal Range (also known as the TRUE BALLISTICRANGE). In one exemplary embodiment, the exemplary icon for BOW togglesbetween bullet calculations (FIGS. 4, 6 and 7) and arrow calculation(FIG. 5), and between ballistic groupings for bullets and arrows, whichare selectable from the menu segments of the exemplary A/B/C menu icon.

One exemplary embodiment of display 900 may also comprise a data display910 including a primary data display section 902 and a secondary datadisplay section 904. Primary data display section 902 may be used tooutput EHR calculations, as indicated by the adjacent exemplary iconlabeled “TBR”. Secondary numerical display 904 may be used to output theLOS range, as indicated by the adjacent exemplary icon labeled “LOS”. Athird data display section 906 may be provided for displaying aninclination angle, measured by an inclinometer sensor 1008 (FIG. 10) ofrangefinder 800. Still further exemplary display sections may beprovided for displaying data representative of a trajectory parameter,such as ballistic path height BP, vertical ballistic drop, ACISinformation, energy, momentum, velocity, etc., at the target range. Inone exemplary embodiment, based on ballistic path height BP or anothertrajectory parameter TP, another display section (not shown) may displaya recommended holdover adjustment in inches, millimeters or mils, at thetarget range or a recommended elevation adjustment in MOA or mils.

A battery power indicator 908 may be included in exemplary display 900for indicating an estimate of the amount of battery power remaining. Oneor more display, segments 909 in the center of the battery powerindicator 908 may be turned of to indicate the remaining battery power.A user-configurable targeting reticle display 910 may also be includedin exemplary display 900 for facilitating aiming of rangefinder 800. Inone exemplary embodiment, exemplary reticle display 910 comprises aplurality of segments that allow exemplary reticle display 910 to bereconfigured in various ways.

FIG. 10 depicts an exemplary block diagram for an exemplary embodimentof rangefinder device 1000 according to the subject matter disclosedherein. In one exemplary embodiment, rangefinder device 1000 could beconfigured similar to the exemplary portable, handheld rangefinderdevice depicted in FIG. 8. In another exemplary embodiment, rangefinderdevice 1000 could be configured to be part of or communicatively coupledto an exemplary telescopic sighting device 1100 depicted in FIG. 11.Rangefinder device 1000 comprises a computer processor or digitalprocessor 1001, such as a microprocessor or digital signal processor(DSP), operatively coupled to laser ranging system 1002, display device1003, and user interface 1004, 1005. Targeting sight 1006 and laserranging system 1002 are aligned relative to each other and are supportedin a common housing 1007, which may include an internal carriage orframe. An inclinometer sensor 1008 is mounted to a support structure inrangefinder device 1000 in alignment with ranging system 1002 andtargeting sight 1006 for measuring the inclination angle θ of the lineof sight (LOS) between a vantage point VP and a target T (FIG. 2). Theballistic calculations described above with reference to FIGS. 1-7 maybe performed by digital processor 1001 of rangefinder device 1000automatically after a laser ranging measurement has been made by rangingsystem 1002.

To facilitate accurate ballistics calculations, digital processor 1001is in communication with inclinometer 1008 and other sensors, such as anelectronic compass 1009, a temperature sensor 1010, abarometer/altimeter sensor 1011, and/or a relative humidity sensor 1012.The data from the sensors may be used as shooting-condition inputs toballistic calculation software operating on digital processor 1001 forperforming the methods described above with reference to FIGS. 1-7. Inone exemplary embodiment, a memory 1013 readable by digital processor1001 stores a software program, sensor data, and/or user-definedsettings, among other information. In one exemplary embodiment, memory1013 may also store data tables including ballistic coefficients forvarious bullets and arrows or groups thereof. In another exemplaryembodiment, memory 1013 may store data tables including ballistic tableswith predicted trajectory parameters for known shooting conditions(including a range of angles) and tables with EHR data (under level-fireconditions) for a range of trajectory parameters. A GPS receiver 1014and antenna 1015 for acquiring geographic location data from GPSsatellite signals may also be included in rangefinder device 1000 inoperative association with digital processor 1001. A signaling module1016, which may include an antenna 1017, may be coupled to digitalprocessor 1001 for transmitting signals representative of ballisticcalculation data calculated by digital processor 1001, such as one ormore trajectory parameters, EHR, elevation adjustments and ACIS holdoveradjustments.

The output of BR, EHR or ACIS information may be displayed via agraphical representation of a corresponding aiming mark of a weaponaiming device reticle or targeting sight. In one exemplary embodiment ofsuch a display method, an aiming mark of the facsimile reticlecorresponding to the output BP, EHR or ACIS data is identified byhighlighting, emphasizing, flashing, coloring, or otherwise changing theappearance of the aiming mark to accomplish a graphical display of therecommended aiming point in relation to the overall reticle pattern. Thegraphical display communicates to the user which of several aiming marksor points on the corresponding riflescope reticle would be recommendedfor use in holdover aiming of a firearm that is separate from therangefinder. In another exemplary embodiment, rangefinder device 1000and a targeting sight are integrated in a common housing with ariflescope or other weapon aiming device, in which the same sightingdevice and reticle display may be used for aiming rangefinder device1000 and for aiming the projectile weapon utilizing a graphical holdoveraiming display. In still another exemplary embodiment, BR, EHR or ACISdata may be transmitted via wires or wirelessly by signaling module 1016and antenna 1017 of rangefinder device 1000 for receipt by a riflescopeor other aiming device, and subsequent display using a graphicaldisplay. Presenting EHR, BP or ACIS information in a graphical displaythat is a facsimile of reticle of a weapon aiming device may help avoidhuman errors that could otherwise result from attempting to manuallyconvert numerical BP, EHR or ACIS data or using it to manually determinewhich of several secondary aiming marks of a riflescope reticle shouldbe used to aim the weapon.

With reference to FIGS. 10 and 11, signaling module 1016 and antenna1017 of rangefinder device 1000 may be configured to send radiofrequency signals to riflescope 1100 mounted on a firearm 1104 or toanother weapon aiming device (not shown). Radio signals may be used towirelessly feed or control a reticle display (not shown) of riflescope1100 viewable through a riflescope eyepiece 1114 for displayingballistics data in the field of view and/or for other purposes. Wirelessdata transmission enables rangefinder device 1000 to be separate fromthe firearm and protected from the effects of recoil and other harshenvironmental conditions to which riflescopes are typically exposed. Forexample, rangefinder device 1000 may be held by a first person, such asa spotter, positioned several meters away from a shooter holding a rifle1104 with a riflescope 1100 that receives data wirelessly fromrangefinder device 1000. Rangefinder device 1000 may also transmit datawirelessly to several different riflescopes or other devicessubstantially simultaneously, allowing a single spotter to provide datato a group of shooters.

In one exemplary embodiment, the signals transmitted by signaling module1016 may include information representative of elevation adjustments tobe made in riflescope 1100 (in minutes of angle (MOA) or fractionalminutes of angle, such as ¼ MOA or ½ MOA) based on ballisticscalculations made by digital processor 1001. Elevation adjustmentsexpressed in MOA or fractions thereof may be displayed in the reticle ofriflescope 1100 and/or be effected via a manual adjustment of anelevation adjustment knob 1120, a motorized elevation adjustmentmechanism, or by controlling or shifting a reticle display or a reticleof riflescope 1100 for offsetting an aiming mark in the amount of aimingadjustment needed, or to show, highlight, or emphasize a fixed orephemeral aiming mark corresponding to the EHR or ACIS informationcalculated by digital processor 1001. The kind of data needed to makesuch an adjustment or aiming mark may depend on whether the riflescopereticle is in the front focal plane or the rear focal plane ofriflescope 1100.

When a recommended elevation adjustment is displayed (in MOA orotherwise) in the reticle display of riflescope 1100, the recommendedelevation adjustment may be updated dynamically as the user manuallyadjusts an elevation setting of riflescope 1100, for example, via anelevation adjustment knob 1120. To enable the recommended elevationadjustment display to be updated dynamically, the elevation adjustmentknob 1120 may include a rotary encoder that provides feedback to adisplay controller of riflescope 1100 or to digital processor 1001.Dynamic updating of the recommended elevation adjustment may enable areticle display to depict the amount of adjustment remaining (e.g.,remaining MOA or clicks of the adjustment knob needed) as the useradjusts elevation, without requiring constant communication betweenriflescope 1100 and rangefinder device 1000 during the elevationadjustment process. Dynamic updating of the remaining adjustment neededmay facilitate operation of rangefinder device 1000 and riflescope 1100sequentially by a single person. In another exemplary embodiment,rangefinder device 1000 may communicate constantly with riflescope 1100,which may allow two people (e.g., a shooter working with a spotter) tomore quickly effect accurate aiming adjustments.

In one exemplary embodiment, signaling module 1016 may include aninfrared transceiver, Bluetooth™ transceiver, or other short-rangelow-power transceiver for communication with a corresponding transceiverof riflescope 1100, for enabling two-way communication while conservingbattery power in rangefinder device 1000 and riflescope 1100. Data forcontrolling a reticle and/or elevation adjustment mechanism 1120 may betransmitted via Bluetooth or other radio-frequency signals. Also,because Bluetooth™ transceivers facilitate two-way communication, therangefinder device 1000 may query riflescope 1100 for a currentelevation adjustment setting, a power adjustment setting, and otherinformation, such as the type of riflescope 1100 and reticle used. Thisdata may then be taken into account in ballistics calculations performedby digital processor 1001. Elevation adjustment and power adjustmentsettings of riflescope 1100 may be determined, for example, by rotaryposition sensor/encoders associated with elevation adjustment knob 1120and power adjustment ring 1130.

In another exemplary embodiment, signaling module 1016 may comprise acable connector plug or socket for establishing a wired connection toriflescope 1100. A wired connection may avoid the need to have delicateelectronics and battery power onboard riflescope 1100. Wired andwireless connections may also be made between signaling module 1016 andother devices, such as bow-sights (including illuminated pin sights andothers), PDAs, laptop computers, remote sensors, data loggers, wirelessdata and telephone networks, and others, for data collection and otherpurposes.

Holdover indication in a riflescope, bow sight, or other optical aimingdevice may be achieved by emphasizing an aiming mark of the sight thatcorresponds to the EHR or ACIS information calculated by rangefinderdevice 1000. In an exemplary ballistic reticle, a primary aiming mark,which may be formed by the intersection or convergence of a primaryvertical aiming line with a primary horizontal aiming line, coincideswith a reference sighted-in range (such as 200 yards horizontal). Asdescribed above and in U.S. Pat. No. 7,603,804 B2 to Zaderey et al.,titled “Ballistic Reticle for Projectile Weapon Aiming Systems andMethod of Aiming,” the disclosure of which is incorporated herein byreference, secondary aiming marks are spaced along a primary verticalaiming line and identify holdover aiming points at which bullet impactwill occur at incremental ranges beyond the sighted-in range.

Although the foregoing disclosed subject matter has been described insome detail for purposes of clarity of understanding, it will beapparent that certain changes and modifications may be practiced thatare within the scope of the appended claims. Accordingly, the presentembodiments are to be considered as illustrative and not restrictive,and the subject matter disclosed herein is not to be limited to thedetails given herein, but may be modified within the scope andequivalents of the appended claims.

What is claimed is:
 1. A method, comprising: determining a line-of-sightrange from a current vantage point to a target that is elevated ordepressed relative to the current vantage point; determining aninclination angle of a line of sight between the current vantage pointand the target; and determining a predicted altitude-compensatedinclined shooting trajectory at the line-of-sight range for apreselected projectile based on a difference in altitude between thecurrent vantage point with respect to sea level and an altitude of asighting-in vantage point with respect to sea level, the sighting-invantage point being a vantage point at which a projectile weaponshooting the preselected projectile was sighted in.
 2. The methodaccording to claim 1, wherein the predicted altitude-compensatedinclined shooting trajectory is based on a bullet path height correctionbetween a bullet path height at the current vantage point with respectto sea level and a bullet path height at the sighting-in vantage pointwith respect to sea level.
 3. The method according to claim 2, whereinthe bullet path height correction is further based on a range distanceto the target from the current vantage point.
 4. The method according toclaim 2, wherein the bullet path height correction is further based onArmy Standard Meteorological Atmosphere information, InternationalStandard Atmosphere information, or actual atmosphere information. 5.The method according to claim 1, wherein the processor is furthercapable of being configured to determine a holdover or a holdundercorresponding to the predicted altitude-compensated inclined shootingtrajectory.
 6. The method according to claim 5, further comprisingdisplaying information based on the holdover or the holdundercorresponding to the predicted altitude-compensated inclined shootingtrajectory.
 7. The method according to claim 1, further comprisingcalculating an angular elevation adjustment for an aiming devicecorresponding to the predicted altitude-compensated inclined shootingtrajectory.
 8. The method according to claim 7, further comprisingdisplaying the angular elevation adjustment representing to thepredicted altitude-compensated inclined shooting trajectory.
 9. Themethod according to claim 7, further comprising communicating to aweapon-aiming device a signal representative of the angular elevationadjustment corresponding to the predicted altitude-compensated inclinedshooting trajectory.
 10. The method according to claim 9, wherein themethod is performed by a weapon-aiming device comprising a rangefinderor a riflescope, the rangefinder or the riflescope comprising anautomatic elevation adjustment mechanism responsive to the signalrepresentative of the angular elevation adjustment.
 11. An articlecomprising: a computer readable medium having stored thereoninstructions that, if executed, result in at least the following:determining a line-of-sight range from a current vantage point to atarget that is elevated or depressed relative to the current vantagepoint; determining an inclination angle of a line of sight between thecurrent vantage point and the target; and determining a predictedaltitude-compensated inclined shooting trajectory at the line-of-sightrange for a preselected projectile based on a difference in altitudebetween the current vantage point with respect to sea level and analtitude of a sighting-in vantage point with respect to sea level, thesighting-in vantage point being a vantage point at which a projectileweapon shooting the preselected projectile was sighted in.
 12. Thearticle according to claim 11, wherein the predictedaltitude-compensated inclined shooting trajectory is based on a bulletpath height correction between a bullet path height at the currentvantage point with respect to sea level and a bullet path height at thesighting-in vantage point with respect to sea level.
 13. The articleaccording to claim 12, wherein the bullet path height correction isfurther based on a range distance to the target from the current vantagepoint.
 14. The article according to claim 12, wherein the bullet pathheight correction is further based on Army Standard MeteorologicalAtmosphere information, International Standard Atmosphere information,or actual atmosphere information.
 15. The article according to claim 11,wherein the processor is further capable of being configured todetermine a holdover or a holdunder corresponding to the predictedaltitude-compensated inclined shooting trajectory.
 16. The articleaccording to claim 15, further comprising displaying information basedon the holdover or the holdunder corresponding to the predictedaltitude-compensated inclined shooting trajectory.
 17. The articleaccording to claim 11, further comprising calculating an angularelevation adjustment for an aiming device corresponding to the predictedaltitude-compensated inclined shooting trajectory.
 18. The articleaccording to claim 17, further comprising displaying the angularelevation adjustment representing to the predicted altitude-compensatedinclined shooting trajectory.
 19. The article according to claim 17,further comprising communicating to a weapon-aiming device a signalrepresentative of the angular elevation adjustment corresponding to thepredicted altitude-compensated inclined shooting trajectory.
 20. Thearticle according to claim 19, wherein the instructions are executed bya weapon-aiming device comprising a rangefinder or a riflescope, therangefinder or the riflescope comprising an automatic elevationadjustment mechanism responsive to the signal representative of theangular elevation adjustment.